Finite Group Databasebeta

← C2×C4×C3⋊(#41⋊C3) #11923173

Subgroups — C2×C4×C3⋊(#41⋊C3) #11923173

Order 1152.

881 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 881 conjugacy classes of subgroups — too many to draw. Only its 151 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3⋊(#41⋊C3)C211yesmaximal, characteristic
2C2×C3×C4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C2×Dic3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4C4×C3⋊(#41⋊C3)C214yesmaximal, direct product factor
5C2×C4×#239C311yesmaximal, characteristic, complemented
6C2×C4×#41⋊C3—31nomaximal
7C22×C3×SL(2,𝔽3)C2211yescharacteristic
8C2×C3⋊(#41⋊C3)C414yesdirect product factor
9C2×C3⋊(#41⋊C3)C2212yes—
10C2×C3×C4×Dic3—41nomaximal
11C3×C4×SL(2,𝔽3)C2212yescomplemented
12Dic3×SL(2,𝔽3)C2212yescomplemented
13C22×#239C2×C311yescharacteristic
14C22×#41⋊C3—31no—
15C2×C3×C4×Q8C2×C311yescharacteristic
16C2×Q8×Dic3C2×C311yescharacteristic
17C4×#239—34no—
18C4×#239C2×C314yescomplemented
19C2×#893—31no—
20C2×#892—31no—
21C2×C4×D12—31no—
22C2×C42×D3—31no—
23C4×#41⋊C3—34no—
24C2×C4×SL(2,𝔽3)—21no—
25C2×C4×SL(2,𝔽3)—31no—
26C2×C4×SL(2,𝔽3)D311yescharacteristic, complemented
27C22×C32×C4—41no—
28C22×C3×Dic3—41no—
29C22×C3×Dic3—41no—
30C2×C3×SL(2,𝔽3)C2×C412yescomplemented
31C2×C3×SL(2,𝔽3)C2311yescharacteristic
32C3⋊(#41⋊C3)C2×C414yesdirect product factor
33C3×C4×Dic3—44no—
34C2×C4×#41—31noSylow
35C22×C3×Q8C22×C311yescharacteristic
36C2×#239—34no—
37C2×#239—32no—
38C2×#239C3×C414yescomplemented
39C2×#239C22×C312yes—
40C22×D12—31no—
41C22×C4×D3—31no—
42C22×C4×D3—31no—
43C22×C4×D3—31no—
44C2×#41⋊C3—34no—
45C2×#41⋊C3—32no—
46C22×SL(2,𝔽3)—21no—
47C22×SL(2,𝔽3)C2×D311yescharacteristic
48C3×C4×Q8—32no—
49C3×C4×Q8C22×C312yes—
50C2×C3×C4⋊C4—31no—
51C2×C3×C42—31no—
52C2×C3×C42—121no—
53Q8×Dic3—32no—
54Q8×Dic3C22×C312yes—
55C2×(C2×D3)⋊C4—32no—
56C2×C4⋊Dic3—31no—
57C2×C4×Dic3—31no—
58C2×C4×Dic3—31no—
59C2×C4×Dic3A411yescharacteristic
60#893—34no—
61#893—34no—
62#892—34no—
63#892—34no—
64C4×D12—34no—
65C4×D12—34no—
66C42×D3—34no—
67C42×D3—34no—
68C4×SL(2,𝔽3)—22no—
69C4×SL(2,𝔽3)—32no—
70C4×SL(2,𝔽3)C2×D312yescomplemented
71C23×C32—41no—
72C2×C32×C4—44no—
73C2×C32×C4—42no—
74C2×C3×Dic3—44no—
75C2×C3×Dic3—44no—
76C2×C3×Dic3—42no—
77C2×C3×Dic3—42no—
78C3×SL(2,𝔽3)C22×C411yescharacteristic, complemented
79C22×#41—31no—
80C4×#41—34no—
81C4×#41—94no—
82C2×C4×Q8—31no—
83C2×C4×Q8C3×D311yescharacteristic
84C2×C4×D4—91no—
85C2×C4×D4—91no—
86C2×#117—91no—
87C22×C42—91no—
88C23×D3—31no—
89C2×C3×Q8—31no—
90C2×C3×Q8—32no—
91C2×C3×Q8C2×C3×C412yes—
92C2×C3×Q8C23×C311yescharacteristic
93C22×C3×C4—31no—
94C22×C3×C4—31no—
95C22×C3×C4—41no—
96C22×C3×C4—81no—
97C22×C3×C4—121no—
98C22×C3×C4—121no—
99C22×C3×C4C2×A411yescharacteristic
100C22×Dic3—31no—
101C22×Dic3C2×A411yescharacteristic
102C22×Dic3C2×A411yescharacteristic
103#239—38no—
104#239—38no—
105#239—34no—
106#239C2×C3×C414yescomplemented
107C2×D12—32no—
108C2×D12—34no—
109C2×D12—32no—
110C2×D12—34no—
111C2×C4×D3—34no—
112C2×C4×D3—34no—
113C2×C4×D3—32no—
114C2×C4×D3—32no—
115C2×C4×D3—34no—
116C2×C4×D3—34no—
117C2×C4×D3—64no—
118C2×C4×D3—64no—
119#41⋊C3—34no—
120C2×SL(2,𝔽3)—22no—
121C2×SL(2,𝔽3)—21no—
122C2×SL(2,𝔽3)C4×D312yescomplemented
123C2×SL(2,𝔽3)C22×D311yescharacteristic
124C3×C4⋊C4—32no—
125C3×C4⋊C4—32no—
126C3×C42—32no—
127C3×C42—32no—
128C3×C42—124no—
129(C2×D3)⋊C4—38no—
130C4⋊Dic3—32no—
131C4⋊Dic3—32no—
132C4×Dic3—34no—
133C4×Dic3—32no—
134C4×Dic3—32no—
135C4×Dic3C2×A414yes—
136C22×C32—42no—
137C22×C32—42no—
138C22×C32—42no—
139C22×C32—41no—
140C32×C4—44no—
141C3×Dic3—44no—
142C3×Dic3—44no—
143C2×#41—34no—
144C2×#41—32no—
145C2×#41—92no—
146C2×#41—94no—
147C22×Q8C2×C3×D311yescharacteristic
148C22×D4—91no—
149C23×C4—91no—
150C23×C4—91no—
151C23×C4—91no—
152C4×Q8—32no—
153C4×Q8—32no—
154C4×Q8—92no—
155C4×Q8C2×C3×D312yes—
156C4×D4—94no—
157C4×D4—94no—
158C4×D4—94no—
159C4×D4—94no—
160#117—94no—
161#117—94no—
162C2×C4⋊C4—31no—
163C2×C4⋊C4—91no—
164C2×C22⋊C4—92no—
165C2×C42—31no—
166C2×C42—31no—
167C2×C42—94no—
168C2×C42—94no—
169C2×C42—91no—
170C2×C42—91no—
171C23×C3—41no—
172C23×C3—81no—
173C23×C3C22×A411yescharacteristic
174C22×D3—34no—
175C22×D3—32no—
176C22×D3—62no—
177C22×D3—62no—
178C3×Q8—32no—
179C3×Q8—31no—
180C3×Q8—32no—
181C3×Q8C22×C3×C411yescharacteristic
182C2×C3×C4—32no—
183C2×C3×C4—32no—
184C2×C3×C4—31no—
185C2×C3×C4—31no—
186C2×C3×C4—32no—
187C2×C3×C4—42no—
188C2×C3×C4—44no—
189C2×C3×C4—62no—
190C2×C3×C4—82no—
191C2×C3×C4—84no—
192C2×C3×C4—122no—
193C2×C3×C4—122no—
194C2×C3×C4—124no—
195C2×C3×C4—124no—
196C2×C3×C4#41⋊C314yescomplemented
197C2×C3×C4C22×A412yes—
198C2×Dic3—32no—
199C2×Dic3—62no—
200C2×Dic3C4×A414yes—
201C2×Dic3#41⋊C314yescomplemented
202C2×Dic3C22×A412yes—
203C2×Dic3C22×A412yes—
204D12—34no—
205D12—34no—
206D12—38no—
207C4×D3—34no—
208C4×D3—38no—
209C4×D3—34no—
210C4×D3—68no—
211C4×D3—68no—
212SL(2,𝔽3)—21no—
213SL(2,𝔽3)C2×C4×D311yescharacteristic, complemented
214C2×C32—41no—
215C2×C32—41no—
216C2×C32—41no—
217C2×C32—44no—
218C24—91no—
219#41—34no—
220#41—98no—
221#41—98no—
222#41—94no—
223C2×Q8—31no—
224C2×Q8—32no—
225C2×Q8C3×C4×D312yes—
226C2×Q8C22×C3×D311yescharacteristic
227C2×D4—92no—
228C2×D4—94no—
229C2×D4—92no—
230C2×D4—94no—
231C22×C4—31no—
232C22×C4—31no—
233C22×C4—31no—
234C22×C4—31no—
235C22×C4—92no—
236C22×C4—92no—
237C22×C4—94no—
238C22×C4—94no—
239C22×C4—94no—
240C22×C4—91no—
241C22×C4—94no—
242C22×C4—184no—
243C22×C4—184no—
244C22×C4D3×A411yescharacteristic
245C4⋊C4—32no—
246C4⋊C4—32no—
247C4⋊C4—92no—
248C4⋊C4—92no—
249C22⋊C4—98no—
250C42—32no—
251C42—32no—
252C42—34no—
253C42—94no—
254C42—92no—
255C42—92no—
256C22×C3—42no—
257C22×C3—42no—
258C22×C3—42no—
259C22×C3—41no—
260C22×C3—82no—
261C22×C3—82no—
262C22×C3—82no—
263C22×C3—81no—
264C22×C3C2×C4×A412yes—
265C22×C3C2×#41⋊C312yes—
266C22×C3C2×#41⋊C312yes—
267C22×C3C23×A411yescharacteristic
268C2×D3—34no—
269C2×D3—62no—
270C2×D3—62no—
271C2×D3—68no—
272C3×C4—31no—
273C3×C4—32no—
274C3×C4—31no—
275C3×C4—44no—
276C3×C4—62no—
277C3×C4—84no—
278C3×C4—124no—
279C3×C4—124no—
280C3×C4C2×#41⋊C314yescomplemented
281Dic3—62no—
282Dic3C2×C4×A414yes—
283Dic3C2×#41⋊C314yescomplemented
284C32—41noSylow
285C23—94no—
286C23—92no—
287C23—182no—
288C23—182no—
289C23C2×D3×A411yescharacteristic
290Q8—32no—
291Q8—31no—
292Q8—32no—
293Q8C2×C3×C4×D311yescharacteristic
294D4—94no—
295D4—94no—
296D4—98no—
297C2×C4—32no—
298C2×C4—32no—
299C2×C4—34no—
300C2×C4—34no—
301C2×C4—31no—
302C2×C4—31no—
303C2×C4—32no—
304C2×C4—32no—
305C2×C4—32no—
306C2×C4—62no—
307C2×C4—94no—
308C2×C4—94no—
309C2×C4—98no—
310C2×C4—92no—
311C2×C4—188no—
312C2×C4—182no—
313C2×C4—188no—
314C2×C4C3⋊(#41⋊C3)14yesdirect product factor
315C2×C4C2×D3×A412yes—
316C2×C3—41no—
317C2×C3—41no—
318C2×C3—41no—
319C2×C3—44no—
320C2×C3—81no—
321C2×C3—81no—
322C2×C3—81no—
323C2×C3—84no—
324C2×C3C4×#41⋊C314yescomplemented
325C2×C3C22×C4×A411yescharacteristic
326C2×C3C22×#41⋊C311yescharacteristic
327C2×C3C22×#41⋊C311yescharacteristic
328D3—64no—
329C22—94no—
330C22—182no—
331C22—182no—
332C22—188no—
333C22C4×D3×A412yes—
334C22C2×C3⋊(#41⋊C3)12yes—
335C22D3×#41⋊C312yes—
336C22C22×D3×A411yescharacteristic
337C4—31no—
338C4—32no—
339C4—31no—
340C4—34no—
341C4—34no—
342C4—62no—
343C4—182no—
344C4C2×C3⋊(#41⋊C3)14yesdirect product factor
345C3—41no—
346C3—81no—
347C3C2×C4×#41⋊C311yescharacteristic, complemented
348C2—184no—
349C2C4×C3⋊(#41⋊C3)14yesdirect product factor
350C2C2×C4×D3×A411yescharacteristic
351C2C22×C3⋊(#41⋊C3)11yescharacteristic
352C2C2×D3×#41⋊C311yescharacteristic
subgroupG/H
center, hypercenterC22×C4D3×A4
commutator, hypocenterC3×Q8C22×C3×C4
FrattiniC22C22×D3×A4
FittingC2×C3×C4×Q8C2×C3
socleC23×C3C22×A4
prime radicalC22×C3⋊(#41⋊C3)C2
cosocleC2×C3×Q8C23×C3
LCSG ▷ C3×Q8
UCSC1 ◁ C22×C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×C3×C4×SL(2,𝔽3)C2semidirect
3. C2×Dic3×SL(2,𝔽3)C2semidirect
4. C4×C3⋊(#41⋊C3)C2direct (N × H)
5. C2×C4×#239C3semidirect
8. C2×C3⋊(#41⋊C3)C4direct (N × H)
11. C3×C4×SL(2,𝔽3)C22semidirect
12. Dic3×SL(2,𝔽3)C22semidirect
18. C4×#239C2×C3semidirect
26. C2×C4×SL(2,𝔽3)D3semidirect
30. C2×C3×SL(2,𝔽3)C2×C4semidirect
32. C3⋊(#41⋊C3)C2×C4direct (N × H)
38. C2×#239C3×C4semidirect
70. C4×SL(2,𝔽3)C2×D3semidirect
78. C3×SL(2,𝔽3)C22×C4semidirect
106. #239C2×C3×C4semidirect
122. C2×SL(2,𝔽3)C4×D3semidirect
196. C2×C3×C4#41⋊C3semidirect
201. C2×Dic3#41⋊C3semidirect
213. SL(2,𝔽3)C2×C4×D3semidirect
280. C3×C4C2×#41⋊C3semidirect
283. Dic3C2×#41⋊C3semidirect
314. C2×C4C3⋊(#41⋊C3)direct (N × H)
324. C2×C3C4×#41⋊C3semidirect
344. C4C2×C3⋊(#41⋊C3)direct (N × H)
347. C3C2×C4×#41⋊C3semidirect
349. C2C4×C3⋊(#41⋊C3)direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×C3⋊(#41⋊C3)2
C2×C3×C4×SL(2,𝔽3)2
C2×Dic3×SL(2,𝔽3)2
C4×C3⋊(#41⋊C3)2
C2×C4×#2393
C2×C4×#41⋊C33
C2×C3×C4×Dic34